AQ 232: Fish Population Dynamics and Stock Assessment

Fish Population Dynamics

Nyamisi Peter

2026-01-21

Mortality (M)

Mortality (M):– The deaths in the population from all causes (natural mortality like disease/predation, plus fishing mortality)

Types of Mortality:

  1. Natural mortality (M) - Disease, predation, starvation, old age

  2. Fishing mortality (F) - Deaths from fishing

  3. Total mortality (Z) - Sum of natural and fishing mortality

    Z = M + F

Mortality (M)…

Mortality characteristics:

  • Often higher at young ages
  • May increase with old age
  • Fishing mortality controllable, natural not
  • Varies by season and location

Mortality Illustration

graph TD
    A["Starting Population<br/>1000 fish"] --> B["Natural Mortality<br/>50 fish die"]
    A --> C["Fishing Mortality<br/>200 fish caught"]
    B --> D["Survivors<br/>750 fish"]
    C --> D
    D --> E["Total Mortality = 25%"]

Population Models

Mathematical Tools for Prediction

Model Basics

What is a Population Model?

  • Mathematical representation of how population changes
  • Uses vital rates (recruitment, growth, mortality)
  • Predicts future population size
  • Basis for harvest decisions

Model Basics…

Simple Population Equation:

\[N_{t+1} = N_t + R_t - M_t - C_t\]

Where:

  • \(N_t\) = population at time t
  • \(R_t\) = recruitment
  • \(M_t\) = natural mortality
  • \(C_t\) = catch

Exponential Growth Model

  • The Exponential Growth Model describes population growth when there are no resource limitations.
  • It assumes that the population grows at a constant rate, leading to exponential increase over time.

Formula:

\[N_t = N_0 e^{rt}\]

Or equivalently:

\[N_t = N_0 \lambda^t\]

Where:

  • \(N_t\) = population size at time t
  • \(N_0\) = initial population size
  • \(r\) = intrinsic rate of increase (growth rate)
  • \(\lambda\) = population growth rate
  • \(t\) = time

What it means:

  • If \(r > 0\): population grows exponentially
  • If \(r = 0\): population stable
  • If \(r < 0\): population declines or shrink

Exponential Growth Model

Assumption:

1. Unlimited resources

  • Food is always available
  • Space is not limiting
  • No environmental constraints
  • Habitat quality doesn’t change

Exponential growth curve:

Time Population Exponential

Exponential Growth Model

Assumption:

2. Constant Growth Rate (r)

  • Growth rate doesn’t change over time
  • Same regardless of population size
  • Environmental conditions stable
  • No density-dependent effects

Exponential growth curve:

Time Population Exponential

Exponential Growth Model

Assumption:

3. No Density Dependence

  • Population size doesn’t affect vital rates
  • Birth rate stays constant regardless of how many fish there are
  • Death rate stays constant
  • Competition doesn’t increase with density

Exponential growth curve:

Time Population Exponential

Exponential Growth Model

Assumption:

4. Closed Population

  • No immigration (fish moving in)
  • No emigration (fish moving out)
  • All changes come from births and deaths only

Exponential growth curve:

Time Population Exponential

Exponential Growth Model

Assumption:

5. Continuous Population

  • Applies to every individual equally
  • No age/size structure effects
  • Simplified view of population

Exponential growth curve:

Time Population Exponential

Exponential Growth Model

Assumption:

6. No Fishing or External Removal

  • All removals are natural deaths only
  • Assumes no harvesting

Exponential growth curve:

Time Population Exponential

Why these assumptions are unrealistic?

Problem with Exponential Model

  • Ignores resource limits

    • Fish DO have limits (food, space, oxygen)

  • Predicts infinite growth

    • Growth rates DO change with density
    • Competition DOES increase when crowded
    • Real populations have age/size structure

  • Unrealistic

    • Fisheries involve harvesting

Logistic Growth Model

  • Logistic growth model is more realistic for fish populations than the exponential growth model.
  • It incorporates the concept of carrying capacity (K), which represents the maximum population size that the environment can sustain indefinitely and density-dependent effects.
Logistic Growth Time Population Logistic (K)

Logistic Growth Model…

Assumption: Growth limited by carrying capacity (K)

Formula:

\[\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)\]

Or in discrete form:

\[N_{t+1} = N_t + rN_t\left(1 - \frac{N_t}{K}\right)\]

Where:

  • \(r\) = intrinsic rate of increase
  • \(K\) = carrying capacity (maximum population)
  • \((1 - N_t/K)\) = density-dependent factor

Logistic Growth Model…

What it means:

  • Growth slows as population approaches K
  • Equilibrium at K
  • More realistic for real populations
Logistic Growth Time Population Logistic (K)

Logistic Growth Dynamics

At different population levels:

  • When N < K: Population grows rapidly
  • When N = K/2: Population growth is fastest
  • When N approaches K: Population growth slows dramatically
  • When N = K: Population is stable (if no fishing)

For Fisheries:

  • MSY often achieved near K/2
  • Overfishing reduces population below optimal levels
Logistic Growth Time Population Logistic (K)

Exponential Growth curve Vs Logistic Growth

Exponential Growth vs. Logistic Growth Time Population Exponential Logistic (K)

Stock-Recruitment Relationships

Connecting Spawners to Offspring

What is Stock-Recruitment?

Definition: The relationship between spawning stock size and recruitment (number of young fish produced)

Key Idea:

  • More spawners = more eggs
  • But relationship may not be linear
  • Environmental variation matters
  • Density-dependent effects

Beverton-Holt Model

  • The Beverton-Holt model is a commonly used stock-recruitment relationship in fisheries science.

Formula:

\[R_t = \frac{aS_t}{1 + bS_t}\]

Where:

  • \(R_t\) = recruitment at time t
  • \(S_t\) = spawning stock size
  • \(a\) = productivity parameter (maximum recruitment potential)
  • \(b\) = density-dependent parameter (strength of density dependence)

Beverton-Holt Model…

Characteristics:

  • Recruitment increases with spawners
  • Recruitment plateaus at high spawning stock (asymptotic)
  • Maximum recruitment = \(a/b\)
  • Good for species with strong density-dependent effects
Beverton-Holt Stock-Recruitment Model Spawning Stock Size (S_t) Recruitment (R_t) Max R = a/b 0 S₁ S₂ S₃ 0 R₁ R₂ R_max Steep increase Plateaus Key Features: • Low S: rapid recruitment • High S: recruitment slows • Asymptotic approach • Never exceeds a/b • Density-dependent • Realistic for many tropical species

How Beverton-Holt Works

At low spawning stock (S_t small):

  • Recruitment increases nearly linearly with spawners
  • Few density-dependent effects
  • Each new spawner adds substantially to recruitment

At high spawning stock (S_t large):

  • Recruitment plateaus (approaches maximum)
  • Density-dependent effects dominate
  • Competition for food/space reduces per-capita recruitment
  • Additional spawners add little to total recruitment

Mathematical insight of Beverton-Holt model

Formula:

\[R_t = \frac{aS_t}{1 + bS_t}\]

  • When \(S_t = 0\): \(R_t = 0\) (no spawners = no recruitment)
  • When \(S_t\) → ∞: \(R_t\) → \(a/b\) (maximum recruitment)

Ricker Model

Formula:

\[R_t = aS_t e^{-bS_t}\]

Where:

  • \(a\) = productivity parameter
  • \(b\) = density-dependent parameter

Ricker Model…

Characteristics:

  • Recruitment increases then decreases
  • Peak recruitment at intermediate spawning stock
  • Decline at very high spawner numbers
  • Good for species (salmon) with cannibalism or crowding effects

Beverton-Holt vs. Ricker

Stock-Recruitment Models Spawning Stock Size Recruitment Beverton-Holt (Asymptotic) Ricker (Hump-shaped)

Key Difference:

  • Beverton-Holt: More spawners = more recruitment (asymptotic)
  • Ricker: Too many spawners can reduce recruitment (hump-shaped)

Variability in Recruitment

graph TD
    A["Spawning Stock Size"] --> B["Environmental<br/>Variation"]
    A --> C["Larval Food<br/>Availability"]
    A --> D["Temperature<br/>& Currents"]
    A --> E["Predation<br/>Pressure"]
    B --> F["Recruitment<br/>Uncertainty"]
    C --> F
    D --> F
    E --> F

Practical Implication:

  • Can’t predict recruitment precisely
  • Must account for uncertainty
  • Risk of recruitment failure

Density-Dependent Effects

How Population Density Matters

What is Density Dependence?

Density Dependence:– When vital rates (growth, survival, reproduction) depend on population density

How it works:

  • High density → increased competition
  • Competition for food, space, mates
  • Reduced growth and survival
  • Lower recruitment

Density Dependence…

Equation:

\[\text{Vital Rate} = f(\text{Population Size})\]

Meaning: Vital Rate is a function of Population Size

  • The vital rate changes depends on how many fish are in the population
  • It is a mathematical function where population size is the input

Density Dependence…

Example 1: Growth Rate

\[Growth~Rate = f(Population~Size)\]

  • Small population: Fish have plenty of food → Fast growth
  • Large population: Competition for food → Slow growth
  • Same species, different growth rate based on population density

Density Dependence…

Example 2: Survival Rate

\[Survival~Rate = f(Population~Size)\]

  • Small population: Low mortality (abundant resources) → High survival
  • Large population: High mortality (starvation, disease) → Low survival

Density Dependence…

Example 3: Reproduction Rate

\[Reproduction~Rate = f(Population~Size)\]

  • Small population: Fish produce many eggs → High recruitment
  • Large population: Fish produce fewer eggs → Low recruitment

Density Dependence…

In contrast, WITHOUT density dependence:

\[Vital~Rate = constant\]

  • Vital Rate does not change with population size

Summary of Density Dependence

Growth:

  • Dense populations grow slower
  • Limited food per fish
  • Smaller fish at same age

Survival:

  • Dense populations have higher mortality
  • Starvation increases
  • Disease spreads easier

Reproduction:

  • Dense populations produce fewer recruits
  • Smaller eggs
  • Lower reproductive success

Density Dependence in Logistic Growth

Density-Dependent Growth Rate Population Size (N/K) Growth Rate Maximum at K/2 K/2 K

Key insight:

  • Growth rate = 0 at K (equilibrium)
  • Maximum growth rate at K/2
  • This is where MSY occurs in logistic model

Thank you!

Questions?